PRZEGLĄD ELEKTROTECHNICZNY 7-2008.pdf

Ukazuje się od 1919 roku

7’08

Organ Stowarzyszenia Elektryków Polskich • Wydawnictwo SIGMA-NOT Sp. z o.o.

Invited paper Tadeusz GLINKA

Politechnika Śląska, Wydział Elektryczny; BOBRME Komel, Katowice

Electric motors with permanent magnets

Abstract. Synchronous motors with permanent magnets placed along the rotor circumference are investigated in the paper. Mathematical models

are given for following motors: PMSM – synchronous motor, PM BLAC – brushless motor with electronic commutator sinusoidally controlled, PM

BLDC - brushless motor with electronic commutator trapezoidally controlled.

Streszczenie. W artykule rozpatruje się silniki synchroniczne z magnesami trwałymi umieszczonymi na obwodzie wirnika. Podaje się modele

matematyczne silnika synchronicznego PMSM, silnika bezszczotkowego z komutatorem elektronicznym sterowanym sinusoidalnie PM BLAC i

silnika bezszczotkowego z komutatorem elektronicznym sterowanym trapezowo PM BLDC. (Silniki elektryczne wzbudzane magnesami

trwałymi).

Key words: synchronous motor, brushless motor, permanent magnets.

Słowa kluczowe : silnik synchroniczny, silnik bezszczotkowy, magnesy trwałe.

1. Motor designs – different variants

Electric motors with magnets located in the rotor are

characterised by highest efficiency of all known and used

electric machines operating with identical electro-

mechanical parameters. Motors with permanent magnets in

the rotor may operate, depending on supply and control, as

- synchronous motors (PMSM – Permanent Magnet

Synchronous Motor),

- brushless dc motors with electronic commutator,

sinusoidally controlled (PM BLAC - Permanent Magnet

Brushless Motor AC)),

- brushless dc motors with electronic commutator,

trapezoidally controlled (PM BLDC - Permanent

Magnet Brushless Motor DC).

Electromechanical properties of each drive are different

–distribution of magnetic field in the slot and control and

supply method constitute deciding factors. Magnetic field

distribution should be adapted to supply conditions. In

synchronous motors and PM BLAC motors induction spatial

distribution should be such that resultant rotational

windings voltage is sinusoidal, while in PM BLDC motors

rotational voltage must be trapezoidal – Fig.2.

150

100

0,0 0 5 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045 0,050

-50

-100

międzyfazowe UV

mię d zyfazowe VW

międzyfazowe WU

-150

time [s]

60,00

40,00

20,00

pasmo 1

pasmo 2

pasmo 3

0,00

0,0050 0,0100 0,0150 0,0200 0,0250 0,0300 0,0350 0,0400 0,0450 0,0500

-20,00

-40,00

-60 00

Fig.1. Electromagnetic circuit of three-phase motor with PM placed

in the rotor

Fig.2. Phase voltages at generator’s terminals during idle run –

a) sinusoidal type control, b) trapezoidal type control

Synchronous motors are supplied with voltage at a

given (set) frequency and rotational speed is changed via

frequency change. Changing the voltage, if frequency and

load torque remain constant ( f = const, T ob = const) will only

affect motor’s reactive power.

In dc brushless motors electronic commutator is built

into the motor in the same way as mechanical commutator

Od Redakcji:

W lipcu bieżącego roku Profesor Tadeusz Glinka obchodzi

jubileusz 70-lecia urodzin (patrz tekst w numerze 6’08).

Z tej okazji Redakcja Przeglądu Elektrotechnicznego zwróciła się z

prośbą do Jubilata o napisanie artykułów odzwierciedlających

ostatnie zainteresowania i osiągnięcia naukowe.

PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 7/2008 1

in standard dc motors. It is supplied with dc voltage, but

current flowing through the windings is variable and its

frequency depends on rotational speed. The speed is set

by changing dc voltage supplying electronic commutator. If

the speed is constant and PWM controller operates so that

windings currents are sinusoidal (this relates to average

current value per one pulse period), then motor is

sinusoidally controlled (BLAC). If current waveforms are

rectangular (trapezoid), then motor is trapezoidally

controlled (BLDC).

2. Synchronous motor

Synchronous motor excited with permanent magnets

should be characterised by sinusoidal voltage waveforms, if

it is correctly designed. It is continuously excited with

magnetomotive force (MMF) and this cannot be varied.

Permanent magnets are glued to the rotor surface (Fig.1 –

this type of design will be investigated in this paper). Since

magnetic relative permeability of permanent magnets is

close to one ( μ≈ 1,03), then for external magnetic field

(generated by armature) the length of magnetic slot is

equal to sum of lengths of air slot and permanent magnets,

calculated along the magnetic lines of the field originated

by armature’s MMF. The outcome of this slot length is that

synchronous reactance is much less than one. Since

permanent magnets ale glued to smooth ferromagnetic

cylinder, synchronous reactances for „d” and „q” axes are

identical and equal to synchronous reactance:

(3)

−

(4)

(5)

- mechanicalpower

(6)

P −

(7)

cos

(8)

cos

sin

−

cos

- reactivepower

(9)

Q =

sin

X S

I A

jX I A

RI A

(1)

XXX

==<

E A

m (P +Q )

U A

There is no damper winding in the rotor and damping

properties of rotor’s ferromagnetic cylinder, with its large

magnetic slot and small electric conductance, are very

weak. Therefore the motor does not generate

asynchronous torque facilitating start-up and it may operate

with inverter supply only, when frequency change ensures

both start-up and speed control.

Steady-state motor operation at ω m = const and T ob =

const ( f = const; U = const) T ob = const, may be illustrated

by space-time diagram - Fig. 3. The time diagram consists

of voltage and current vectors for a chosen phase „A”,

rotating in relation to time axis „t”; space diagram consists

of excitation MMF Θ PM and armature MMF Θ a rotating in

relation to „A” phase axis, and time axis „t” overlaps „A”

phase axis. This is a standard schoolbook diagram [2],

which helps to interpret motor operation from a physical

viewpoint.

Basing on the diagram shown in Fig. 3, mathematical

model of the motor’s steady-state operation may be

defined. The equations may also be used to analyse quasi-

steady states, when frequency f , voltage U or load torque

T ob. change slowly.

The assumptions for the model are:

- motor’s electromagnetic circuit is symmetrical

- magnetic circuit is linear; this is true since excitation

flux is constant due to permanent magnets and

magnetic slot is large

- reactances in d and q axes X s = X d = X q , are equal

- iron losses are equal to zero

- cogging torques are neglected.

Motor equations:

- phase voltages (for phase A)

oś fazy "A"

oś czasu "t"

ω e

jX S

E A

U A

I A

oś N-S

ω m

Fig.3. Space-time diagram characterising steady-state operation of

synchronous motor: a) equivalent scheme for phase A, b) vector

diagram of phase A voltage and current, which rotate in relation to

„ t ” time axis and vector diagram of θ PM MMFs , θ a – armature MMF , θ

- resultant MMF for forces rotating in relation to phase A axis

Phase rotational voltage E f is the electromotive force (EMF)

induced at idle run ( I = 0) at angular speed ω m

(10)

E =

1000

ω m

1000

(2)

−

and line voltage E 1000 is one of motor constants shown

either on terminal board or given in a motor catalogue. This

is equal to line voltage induced during idle run ( I = 0) at

rotational speed n = 1000 rpm ( ω 1000 = 104.6 1/s)

- torque equations

2 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 7/2008

The set of equations (2-10), which contains 9 following

unknowns: E f , I, ω m , T em , P 2 , P 1 , φ , δ , Q will be solvable if

drive and motor parameters are known ( m, R, X s , J, p ), and

supply and load conditions are known as well (( U, f, T ob ).

and the load torque as

(12)

T e

T e max

Fig.7. Measurement scheme for testing synchronous motor M

Fig.4. Synchronous torque angle characteristics at f = const,

U = const, P = var

3. DC motor with electronic commutation

Dc motor with mechanical commutator is characterised by

perpendicular vector layout: one vector is excitation flux Φ

and the other is armature’s MMF Θ a - see Fig.8. This

perpendicularity is permanent, since it results from motor

construction: this is electrical angle α e = p α m between

excitation poles axis and brushes axis.

T em max

U n

Fig.5. PMSM maximum torque vs. voltage at f = const

Basing on the above set of equations, motor

electromechanical properties may be derived:

- torque-angle characteristic T e = f ( δ ) at U = const,

f = const (Fig.4),

- maximum torque vs. voltage characteristic

T e max = f (U) at f = const (Fig.5),

P ,Q

P 1

Fig.8. Excitation flux Φ and armature MMF Θ a vectors -

perpendicular to each other in dc machine with mechanical

commutator

Q 1

The construction of dc motor with electronic commutator

is reversed – Fig. 9 (with the exception of low power motors

used in fans). Excitation flux Φ vector rotates with angular

speed ω m . The angle between „A” phase axis and the axis

of flux Φ vector is a function of time:

(13)

T n

Fig.6. Active power P 1 and reactive power Q 1 vs. load torque at f n =

const and U n = const

- active power P 1 and reactive power Q as a function of

load torque P 1 ; Q = f ( T ob ) ar U = const, f = const –

Fig.6.

All of the above characteristics may be obtained in a

measurement scheme shown in Fig.7.

If motor M and generator G are identical, then there is

no need to measure the load torque T ob (and torque

measurements are usually troublesome).

Power P 2 may be calculated as the arithmetic mean of

P 1 and P 3 powers:

Armature’s MMF Θ a rotates with mechanical speed ω a .

The angle between MMF Θ a and „A” phase axis is a

function of time:

(14)

α +

In order to operate as a dc motor (Fig.8), the angular

speeds must be equal ω m = ω a , and the angles should be

related as follows:

(11)

(15)

−

PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 7/2008 3

Position of Θ a vector in relation to „A” phase axis is

forced by instantaneous values of windings’ currents i A , i B ,

i C . These values should be modified in such a way by

electronic commutator K, that MMF vector Θ a is kept

perpendicular to flux vector Φ . Therefore it is indispensable

to measure angle α m continuously, that is position of flux Φ

in relation to „A” phase axis. If the motor operates in

steady-state, i.e. ω m = const, then i A , i B , i C cirrents vary

sinusoidally. Sinusoidal current waveforms (average

values) are ensured by commutator K. The commutator

valves (1-6) are controlled with PWM sinusoidal algorithm.

This type of motor control results in the so-called dc motor

with electronic commutator sinusoidally controlled (BLAC -

brushless ac motor).

Encoders may be used for continuous angle α m

measurement; however, their construction is complex and

they are expensive, coupling with the motor shaft is also

somewhat difficult. That is why other type of motor control,

the so-called trapezoidal control, has been developed.

i A

1 π

3 π

i B

1 π

3 π

i C

1 π

3 π

ω m

Θ a

oś fazy "A"

Fig.10 a – current waveforms (trapezoidal) in phase windings

A,B,C, b- Φ and θ e vector positions in relation to phase winding A

axis

Fig.9. Electronic commutator motor control: a) location of Φ i Θ a

vectors in relation to phase axes, b) commutator, c) transistor

switching histogram for electronic commutator K

Trapezoidal control is based on discrete (non-

continuous) measurement of α m angle. If flux vector (+ Φ ,

that is N-pole axis) is positioned directly against

measurement point (e.g. „A” phase), then electronic

commutator’s positive valve (1) is switched on – Fig.9.

Phases B and C are controlled similarly. Valve (1) of „A”

phase is switched off when valve (3) of „B” phase is

4 PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 7/2008

switched on, and this valve is switched off when valve (5) of

„C”phase is switched on and so forth. Phases B and C

operate in the same way. If flux vector (- Φ , that is S-pole

axis) is positioned directly against measurement point

related to „A”phase , then electronic commutator’s

negative valve (2) is switched on, in turn it is switched off

when valve (4) of B phase is switched on and so forth.

Electronic commutator is included in the mathematical

model of the motor, these elements constitute an integral

entity, same as in dc motor with mechanical commutator.

Electrical parameters such as voltage u , current i and EMF

e are read at the commutator output. This physical model,

composed of motor and commutator, is equivalent to dc

motor with three commutator bars K = 3 [1] – Fig.12.

Assumptions:

- phase windings are star-connected,

- cogging torque is neglected,

- current switching from phase to phase is conducted

without any perturbations (this means that ω m = const),

switchings are symmetrical and occur every time that

Φ vector moves by angle

2 π

3 π

α m

m = ,

1 π

α a

- when phase is active, rotational voltage is constant if

ω m = const.

Motor equations for motor driving mechanical system with

load torque T ob and inertia load J :

- voltage equation

m t

3 π

(16)

2 +

Fig.11. Changing angle α e at α m = ω m t

- torque equation

With this type of motor control, windings’ currents

waveforms i A , i B , i C assume shapes similar to trapezium

(this is due to winding’s electromagnetic time constant).

The key factor of this control is discrete (step-by-step)

movement of MMF Θ a . Armature’s MMF assumes 6

characteristic positions (for p = 1) - see Fig.10. The step

angle α Θ is equal to π⁄ 3 and subsequent steps oscillate in

relation to straight line perpendicular to Φ flux vector. Each

dislocation of Φ vector by angle

(17)

ω =−

Tt T

()

α = causes a

In dc motor with PM excitation an important parameter

defined by the manufacturer is rotational voltage at idle run

( i = 0) and rotational speed n = 1000 rpm. This is the so-

called E 1000 parameter.

On the basis of E 1000 it is calculated that:

discrete change in ampere-turns vector Θ a by π /3 – see

Fig.11. This motor is called dc motor with electronic

commutator trapezoidally controlled (BLDC – dc brushless

motor). It is characterised by simple rotor angle position

sensors, these are usually photoelectric or Hall effect

sensors.

(18)

e =

1000 ω

ω m

1000

(19)

3.1. Mathematical model of dc electric motor with electronic

commutation and trapezoidal control

It has been remarked previously, that dc motor with

electronic commutator and trapezoidal control should be

characterised by trapezoidal rotational voltage – Fig.2. If

this is not true, then during voltage switching to next phase

current surges occur. These generate electromagnetic

torque impulses and, in turn, additional component of

rotational speed, which is decidedly not a desired effect. In

order to avoid these phenomena, motor should be current-

controlled.

and

ω = 1/s,

1000

104.6

where R, L – resistance and inductance of one winding

phase.

Fig.13. Rotational speed ω m vs. time for PMDCBMTC at U(t) = U ⋅

1 (t) and T ob =0

Fig.12. Equivalent scheme of dc electronic commutator motor with

trapezoidal-type control

This simple set of equations, if motor parameters R, L,

J, E 1000 are known, if supply voltage u is set and load torque

T ob. is given, makes it possible to determine static and

dynamic motor properties such as e.g. motor reaction to a

unit step voltage supply u = U o 1 (t) at T ob. = 0 – Fig.13.

PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 84 NR 7/2008 5

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